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Expected value of a wiener process on an infinite time horizon with a barrier
Say I have a wiener process with $X(0) = X_0>0$ and the dynamics
\begin{equation}
dX(t) =
\begin{cases}
-\mu dt + \sigma X(t) dW(t)^{\mathbb{Q}} & \mathrm{for\ } X(t)>0\\
0 & \mathrm{otherwise}\\
\en …